Here's what I think. Let:
[tex]g(x) := \int f(x)dx \quad \mbox{and} \quad h(x):= \int f(x)dx[/tex]. Then, you do not have g(x) - h(x) = 0, you will have g(x) - h(x) = C, where C is some constant.
So here's the same, you can say that:
[tex]\int \frac{dx}{x} - \int \frac{dx}{x} = C[/tex], where C is some constant.
So it's not a contradiction...
Viet Dao,